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An Apéry-like difference equation for Catalan’s constant. (English) Zbl 1093.11075
Summary: Applying Zeilberger’s algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan’s constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan’s constant as well as a new continued-fraction expansion for it. Similar arguments are put forward to deduce a second-order difference equation and a new continued fraction for $$\zeta(4)=\pi^4/90$$.

##### MSC:
 11Y60 Evaluation of number-theoretic constants 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) 39A05 General theory of difference equations
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